---
title: "Numerical dispersion, stability; four-point implicit (500 m)"
subject: "Computational Techniques in Civil Engineering"
chapter: 5
key: "semester_8/computational_techniques_in_civil_engineering:numerical-dispersion-four-point-j"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/numerical-dispersion-four-point-j
site: BCT Engineering
---

# Numerical dispersion, stability; four-point implicit (500 m)

Computational Techniques in Civil Engineering · Chapter 5: Finite difference method

- Asked in: 2079 Jestha, 4+4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/numerical-dispersion-four-point-j
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

Describe numerical dispersion, diffusion, and stability of finite difference schemes. The value of flow rate $Q$ at four points in the space-time grid is shown in the figure below. Determine the value of first-order derivatives $\partial Q/\partial t$ and $\partial Q/\partial x$ by using the four-point implicit method. Given: $\Delta t = 1$ hour, $\Delta x = 500$ m and $\theta = 0.55$.

[Figure: space-time grid with time levels n and n+1 and distance points i and i+1; $Q$ at (i, n+1) $= 75$ m$^3$/s, at (i+1, n+1) $= 72$ m$^3$/s, at (i, n) $= 73$ m$^3$/s, at (i+1, n) $= 69$ m$^3$/s]

## Short answer

Numerical diffusion smears fronts; dispersion causes oscillations; explicit schemes require Courant number ≤1. Using four-point implicit (θ=0.55), Δt=1 h, Δx=500 m, ∂Q/∂t = 6.94×10^-4 m^3/s^2 and ∂Q/∂x = -0.00690 m^2/s.

## Answer

#### Numerical diffusion
The leading truncation error of an odd-order scheme (e.g. first-order upwind) behaves like a physical diffusion term. For $Q_t+cQ_x=0$ with backward-space, forward-time:
$$
Q_t+cQ_x=\frac{c\Delta x}{2}(1-C)\,Q_{xx}+\dots
$$
The solution is smeared: peaks are lowered and sharp fronts spread out, although the true equation has no diffusion.

#### Numerical dispersion
For second-order (central) schemes the leading error term has an odd derivative, $\propto Q_{xxx}$. Different wavelengths then travel at different speeds, so a sharp front produces spurious oscillations (ripples ahead of or behind the front). It is a phase error, with little change in amplitude.

#### Stability
A scheme is stable if errors (round-off or truncation) do not grow from step to step. For explicit schemes, the von Neumann or Courant-Friedrichs-Lewy condition requires
$$
C=\frac{c\,\Delta t}{\Delta x}\le1
$$
Implicit schemes (and the four-point scheme with $\theta\ge0.5$) are unconditionally stable, but a large $\Delta t$ still reduces accuracy.

#### Four-point implicit method
$$
\frac{\partial Q}{\partial t}\approx\frac{(Q_i^{n+1}-Q_i^n)+(Q_{i+1}^{n+1}-Q_{i+1}^n)}{2\Delta t}
$$
$$
\frac{\partial Q}{\partial x}\approx\theta\,\frac{Q_{i+1}^{n+1}-Q_i^{n+1}}{\Delta x}+(1-\theta)\,\frac{Q_{i+1}^n-Q_i^n}{\Delta x}
$$
Data: $Q_i^{n+1}=75$, $Q_{i+1}^{n+1}=72$, $Q_i^n=73$, $Q_{i+1}^n=69$; $\Delta t=3600$ s, $\Delta x=500$ m, $\theta=0.55$.

**Time derivative**
$$
\frac{\partial Q}{\partial t}=\frac{(75-73)+(72-69)}{2\times3600}=\frac{5}{7200}=6.9444\times10^{-4}\ \text{m}^3/\text{s}^2
$$
**Space derivative**
$$
\frac{\partial Q}{\partial x}=0.55\,\frac{72-75}{500}+0.45\,\frac{69-73}{500}=-0.00330+(-0.00360)=-0.00690\ \text{m}^3/\text{s per m}
$$
**Answer:** $\partial Q/\partial t=6.9444\times10^{-4}$ m$^3$/s$^2$ and $\partial Q/\partial x=-0.00690$ m$^2$/s.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the effect of numerical diffusion in a first-order scheme?
   A. Spurious oscillations ahead of the front
   B. The wave speed doubles
   C. Sharp fronts are smeared and peaks lowered
   D. The scheme becomes unstable
   Answer: C. The leading truncation term acts like a Q_xx diffusion term, smearing the solution. (Key point: Numerical diffusion)

2. Which describes numerical dispersion in second-order (central) schemes?
   A. Different wavelengths travel at different speeds, giving oscillations near fronts
   B. Peaks are lowered by smearing
   C. All waves travel at the same speed
   D. Errors grow without bound
   Answer: A. The leading error has a Q_xxx term, a phase error that creates ripples near sharp fronts. (Key point: Numerical dispersion)

3. With the data given, what is dQ/dt by the four-point method?
   A. 5/7200 = 6.94e-4 m^3/s^2
   B. 5/3600 = 1.39e-3 m^3/s^2
   C. 2/7200
   D. 9/7200
   Answer: A. dQ/dt = [(75 - 73) + (72 - 69)]/(2 x 3600) = 5/7200. (Key point: Four-point time derivative)

4. What is dQ/dx with theta = 0.55?
   A. -0.00330
   B. -0.00690
   C. -0.00720
   D. -0.00360
   Answer: B. 0.55(72 - 75)/500 + 0.45(69 - 73)/500 = -0.0033 - 0.0036 = -0.0069. (Key point: Four-point space derivative)

## Close variants set in other exams

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